Cosmic web analysis and information theorysome recent results
Florent LeclercqInstitute of Cosmology and Gravitation, University of Portsmouth
January 5th, 2016
In collaboration with:Jens Jasche (ExC Universe, Garching), Guilhem Lavaux (IAP),
Will Percival (ICG), Benjamin Wandelt (IAP/U. Illinois)
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Uncertainty quantification
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Can we uncertainty
quantification to ?
Uncertainty quantification is crucial!
Yes, and this is what yields a connection
with !
Cosmic web classification procedures
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• The :
uses the sign of : eigenvalues of the tidal field tensor, Hessian of the gravitational potential:
Hahn et al. 2007, arXiv:astro-ph/0610280
void, sheet, filament, cluster?
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Final conditions
FL, Jasche & Wandelt 2015, arXiv:1502.02690
T-web structures inferred by BORG
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Initial conditions
FL, Jasche & Wandelt 2015, arXiv:1502.02690
T-web structures inferred by BORG
What is the information content of these maps?
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in shannons (Sh)
Initial conditionsFinal conditions
FL, Jasche & Wandelt 2015, arXiv:1502.02690
Shannon entropy
How much did the data surprise us?
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Initial conditionsFinal conditions
in Sh
FL, Jasche & Wandelt 2015, arXiv:1502.02690
information gain a.k.a. relative entropy or Kullback-Leibler divergence posterior/prior
A decision rule for structure classification
• Space of “input features”:
• Space of “actions”:
• A problem of :one should take the action that maximizes the utility
• How to write down the gain functions?
8FL, Jasche & Wandelt 2015, arXiv:1503.00730
• One proposal:
• Without data, the expected utility is
• With , it’s a fair game always play
“ ” of the LSS
• Values represent an aversion for risk
increasingly “ ” of the LSS
Gambling with the Universe
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“Winning”
“Loosing”
“Not playing”
“Playing the game”
“Not playing the game”
FL, Jasche & Wandelt 2015, arXiv:1503.00730
voidssheetsfilamentsclusters
1.74
7.08
3.83
41.67(T-web, final conditions)
Playing the game…
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Final conditions
voids
sheets
filaments
clusters
undecided
Initial conditions
FL, Jasche & Wandelt 2015, arXiv:1503.00730
Cosmic web classification procedures
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• The :
uses the sign of : eigenvalues of the tidal field tensor, Hessian of the gravitational potential:
• :
uses the sign of : eigenvalues of the shear of the Lagrangian displacement field:
• :
uses the dark matter “phase-space sheet” (number of orthogonal axes along which there is shell-crossing)
Hahn et al. 2007, arXiv:astro-ph/0610280
Lavaux & Wandelt 2010, arXiv:0906.4101
Falck, Neyrinck & Szalay 2012, arXiv:1201.2353
Lagrangianclassifiers
void, sheet, filament, cluster?
now usable in real data!
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Comparing classifiersFi
lam
ents
Void
s
FL, Jasche & Wandelt 2015, arXiv:1502.02690
FL, Jasche, Lavaux & Wandelt 2016, arXiv:1601.00093
How similar are different classifications?
14FL, Lavaux, Jasche & Wandelt, in prep.
Jensen-Shannon divergence
(more about the Jensen-Shannon divergence later)
Which is the best classifier?
• Can we extend the decision problem to the space of classifiers?
• As before, the idea is to maximize a utility function
• An important notion: the between two random variables
• Property:
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Mutual information is the expectation of the Kullback-Leibler divergence of the conditional from the unconditional distribution.
FL, Lavaux, Jasche & Wandelt, in prep.
1. Utility for parameter inference:cosmic web analysis• In analogy with the formalism of
: maximize the for cosmic web maps
16FL, Lavaux, Jasche & Wandelt, in prep.
classification data
2. Utility for model selection:dark energy equation of state
• For example, consider three dark energy models with
• The between posterior predictive distributions can be used as an approximate
• In analogy:
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model classifier mixture distribution
Vanlier et al. 2014, BMC Syst Biol 8, 20 (2014)
FL, Lavaux, Jasche & Wandelt, in prep.
3. Utility for prediction of new data:galaxy colors
• Maximize the for some new quantity
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predicted data classification
FL, Lavaux, Jasche & Wandelt, in prep.
3. Utility for prediction of new data:galaxy colors
• How to compute the information gain?
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parent entropy:
child2 entropy:
child1 entropy:
weighted average entropy of children:
information gain for this split:
3. Utility for prediction of new data:galaxy colors
• A problem!
• 3 = classifications (T-web, DIVA, ORIGAMI) with
• 4 (void, sheet, filament, cluster)
• 2 (red, blue)
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X Y Z C
3 2 3 I
3 1 3 I
2 2 0 II
3 1 0 II
no gain: worst best!
X=3
Y=0
Y=1
Y=2
Y=3
Z=0
Z=1
Z=2
Z=3
X=0
X=1
X=2
FL, Lavaux, Jasche & Wandelt, in prep.
Conclusions
• Thanks to , the can be described using various classifiers
• Probabilistic analysis of the cosmic web yields a data-supported
offers a framework to classify structures in the presence of uncertainty
• It is now possible to !
• The decision problem can be extended to the , with utility functions depending on the desired use
(Some numerical results for classifier utilities in the upcoming paper)
21FL, Lavaux, Jasche & Wandelt, in prep.
FL, Jasche, Lavaux & Wandelt 2016, arXiv:1601.00093
FL, Jasche & Wandelt 2015, arXiv:1503.00730
